1.3 More on Functions and craphs \[ \text { Evaluate the piecewise function at the given values of the independent variable. } \] \[ (x)=\left\{\begin{array}{ll}\frac{x^{2}-36}{x-6} & \text { if } x \neq 6 \\ 2 & \text { if } x=6\end{array}\right. \] \( \begin{array}{lll}\text { (a) } h(3) & \text { (b) } h(0) & \text { (c) } h(6)\end{array} \)

To evaluate the piecewise function \( h(x) \) at the given values, let's first understand the function and then substitute the specific values. ### Given Function: \[ h(x) = \begin{cases} \displaystyle \frac{x^{2} - 36}{x - 6} & \text{if } x \neq 6 \\ 2 & \text{if } x = 6 \end{cases} \] ### Simplifying the Function for \( x \neq 6 \): For \( x \neq 6 \), \[ \frac{x^{2} - 36}{x - 6} = \frac{(x - 6)(x + 6)}{x - 6} \] As long as \( x \neq 6 \), you can cancel out the \( (x - 6) \) terms: \[ h(x) = x + 6 \] ### Evaluations: Now, let's evaluate \( h(x) \) at the specified values. #### (a) \( h(3) \) Since \( 3 \neq 6 \), use the simplified expression: \[ h(3) = 3 + 6 = 9 \] #### (b) \( h(0) \) Since \( 0 \neq 6 \), use the simplified expression: \[ h(0) = 0 + 6 = 6 \] #### (c) \( h(6) \) Here, \( x = 6 \), so we use the piecewise definition directly: \[ h(6) = 2 \] ### Summary of Results: - **(a) \( h(3) = 9 \)** - **(b) \( h(0) = 6 \)** - **(c) \( h(6) = 2 \)**